Asymptotic analysis at any order of Helmholtz's problem in a corner with a thin layer: an algebraic approach

TL;DR

This paper develops an algebraic method to derive high-order asymptotic expansions for the Helmholtz problem in a corner with a thin layer, addressing singular behaviors and providing rigorous error estimates.

Contribution

It introduces an algebraic approach to match asymptotic expansions at any order for Helmholtz problems in corners with thin layers, handling singularities systematically.

Findings

Asymptotic expansions are valid at any order with rigorous error bounds.

Matching conditions are reformulated as algebraic relations involving singular behaviors.

Coefficients in the expansions can be computed analytically.

Abstract

We consider the Helmholtz equation in an angular sector partially covered by a homogeneous layer of small thickness, denoted $\varepsilon$. We propose in this work an asymptotic expansion of the solution with respect to $\varepsilon$ at any order. This is done using matched asymptotic expansion, which consists here in introducing different asymptotic expansions of the solution in three subdomains: the vicinity of the corner, the layer and the rest of the domain. These expansions are linked through matching conditions. The presence of the corner makes these matching conditions delicate to derive because the fields have singular behaviors. Our approach is to reformulate these matching conditions purely algebraically by writing all asymptotic expansions as formal series. By using algebraic calculus we reduce the matching conditions to scalar relations linking the singular behaviors of the fields. These relations have a convolutive structure and involve some coefficients that can be computed analytically. Our asymptotic expansion is justified rigorously with error estimates.